Luminescent coupling is a characteristic of multi-junction solar cells which has often been neglected in models of their performance. The effect describes the absorption of light emitted from a higher band gap semiconductor by a lower band gap semiconductor. In this way, light which might have been lost can be utilized for current generation. We present a framework for modeling this effect in both planar layer stacks and devices with periodic nanostructuring. As a case study, we evaluate how luminescent coupling is affected by the inclusion of nanostructuring in a perovskite silicon tandem solar cell. We find that nanostructuring, while reducing the reflection loss for tandem solar cells also reduces the luminescent coupling, allowing more light to be emitted to the surroundings, when compared to planar devices. This highlights the need to include modeling of this effect into optimization schemes in order to find the trade-off between these two effects.
Modelling the scattering of focused, coherent light by nano-scale structures is oftentimes used to reconstruct or infer geometrical or material properties of structures under investigation in optical scatterometry. This comprises both periodic and aperiodic nano-structures. Coherent Fourier scatterometry with focused light exploits the diffraction pattern formed by the nano-structures in Fourier plane. While the scattering of a focused beam by a spatially isolated scatterer is a standard modelling task for state-of-the art electromagnetic solvers based, e.g., on the finite element method, the case of periodically structured samples is more involved. In particular when the focused light covers several grating periods of as it is commonly the case. We will present a coherent illumination model for scattering of focused beams such as Gaussian- and Besselbeams by periodic structures such as line gratings. The model allows to take into account optical wavefront aberrations in optical systems used for both, the illumination and detection of the scattered fields. We compare the model with strategies implemented on large-scale super-cells and inverse Floquet-transform strategies to superimpose both near- and far fields coherently.
In this contribution, we present a technique for the determination of optical aberrations, which is based on measurements of the point spread function and a Bayesian optimization of rigorous simulations. The measuring system is a UV-microscope in a reflected light configuration with a 200x magnification, unpolarized light, and an illumination and imaging NA of 0.44 and 0.55, respectively. The PSF is measured by imaging a small quadratic chrome dot (side length ≈ 180 nm) on a glass substrate. We investigate the impact of different adjustment states, different dot locations and different optical microscopes.
KEYWORDS: Nanostructures, Electron microscopy, Scattering, Diffraction limit, Diffraction, Optical microscopy, Near field optics, Near field, Multilayers, Modeling
Dimensional optical microscopy allows for the rapid inspection of devices at the cost of limited accuracy. Introducing a model-based approach that includes diffraction effects allows for increased accuracies. The model needs to be efficient and accurate to evaluate the measurements in an acceptable time frame.
We present an overview of the illumination model and different incidence-pupil sampling techniques. Furthermore, we will demonstrate strategies for efficiently calculating the near-field scattering response from structures using the finite element method.
Using these aspects, we demonstrate a significant increase in the accuracy of dimensional estimates for a range of structures.
Manual optimization of experimental parameters can quickly become too complex and time-consuming if more than a few correlated parameters need to be adjusted. We discuss automating this process using Bayesian optimization. This machine learning-based method is particularly suitable because it can handle noisy measurements, performs a global search and requires relatively few experimental runs. We discuss the efficient, scalable implementation of Bayesian optimization, present practical applications for tuning experimental parameters, and compare it with other local and global heuristic methods to show its application range.
In many industrial sectors, dimensional microscopy enables non-destructive and rapid inspection of manufacturing processes. However, wave-optical effects and imaging errors of the optical system limit the accuracy. With modelbased approaches it is possible to measure the physical position of edges and corners with submicron uncertainty. This requires an accurate model for phase aberrations of the optical system. We present a method to model and quantify those phase aberrations by an efficient inverse measurement.
This Conference Presentation, “Numerical optimization of resonant photonic devices,” was recorded at SPIE Optical Metrology 2019, held in Munich, Germany.
Optical scatterometry is a method to measure the size and shape of periodic micro- or nanostructures on surfaces. For this purpose the geometry parameters of the structures are obtained by reproducing experimental measurement results through numerical simulations. We compare the performance of Bayesian optimization to different local minimization algorithms for this numerical optimization problem. Bayesian optimization uses Gaussian-process regression to find promising parameter values. We examine how pre-computed simulation results can be used to train the Gaussian process and to accelerate the optimization.
For optimizing specific functionalities of optical components which include structures on a micrometer or nanometer scale, typically high-dimensional optimization problems have to be solved. We use Gaussian process regression to this aim. Gaussian processes can be viewed as machine-learning algorithms where results from evaluations at specific points in the parameter space (training data) are used to predict values and their uncertainty in the full parameter space. The forward-problem (evaluation at a given point in parameter space) requires to rigorously solve Maxwell’s equations, i.e. to compute light propagation in a specific setup. We use our finite-element method (FEM) implementation JCMsuite to this aim. The general framework of FEM allows to employ adaptive numerical resolution and accurate geometry modelling for arbitrary shapes. We discuss application of Bayesian optimization for the inverse problem in parameter retrieval from scatterometric data.
KEYWORDS: Vertical cavity surface emitting lasers, Oxides, Finite element methods, Computer simulations, Near field optics, Maxwell's equations, 3D modeling
Open resonant optical devices such as an oxide-confined vertical cavity surface emitting laser (VCSEL) can be characterized by a quasi-normal mode (QNM) expansion. In contrast to eigenmodes of a closed system, QNMs exhibit an exponential divergence in the exterior of the device and are no longer normalizable. This behavior renders the mathematical treatment and physical understanding very challenging. As an alternative we investigate the constant-flux mode (CFM) expansion which avoids the divergence in the exterior domain. Besides numerical studies, we present results for different oxide aperture sizes and positions inside the investigated VCSEL. Here, we apply CFMs to describe the impact on the resonance wavelength and on the mode profile.
We present a Newton-like method to solve inverse problems and to quantify parameter uncertainties. We apply
the method to parameter reconstruction in optical scatterometry, where we take into account a priori information
and measurement uncertainties using a Bayesian approach. Further, we discuss the influence of numerical
accuracy on the reconstruction result.
A novel design for high brightness planar technology light-emitting diodes (LEDs) and LED on-wafer arrays on absorbing substrates is proposed. The design integrates features of passive dielectric cavity deposited on top of an oxide– semiconductor distributed Bragg reflector (DBR), the p–n junction with a light emitting region is introduced into the top semiconductor λ/4 DBR period. A multilayer dielectric structure containing a cavity layer and dielectric DBRs is further processed by etching into a micrometer–scale pattern. An oxide–confined aperture is further amended for current and light confinement. We study the impact of the placement of the active region into the maximum or minimum of the optical field intensity and study an impact of the active region positioning on light extraction efficiency. We also study an etching profile composed of symmetric rings in the etched passive cavity over the light emitting area. The bottom semiconductor is an AlGaAs–AlAs multilayer DBR selectively oxidized with the conversion of the AlAs layers into AlOx to increase the stopband width preventing the light from entering the semiconductor substrate. The approach allows to achieve very high light extraction efficiency in a narrow vertical angle keeping the reasonable thermal and current conductivity properties. As an example, a micro-LED structure has been modeled with AlGaAs-AlAs or AlGaAs-AlOx DBRs and an active region based on InGaAlP quantum well(s) emitting in the orange spectral range at ~610 nm. A passive dielectric SiO2 cavity is confined by dielectric Ta2O5/SiO2 and AlGaAs-AlOx DBRs. Cylindrically–symmetric structures with multiple ring patterns are modeled. It is demonstrated that the extraction coefficient of light to the air can be increased from 1.3% up to above 90% in a narrow vertical angle (full width at half maximum (FWHM) below 20°). For very small oxide–confined apertures ~100nm the narrowing of the FWHM for light extraction can be reduced down to 5°. Consequently high efficiency high brightness arrays of micro-LEDs becomes possible. For single emitters the approach is particularly interesting for oscillator strength engineering allowing high speed data transmission and for single photonics applying single quantum dot (QD) emitters and allowing >90% coupling of the emission into single mode fiber. We also note that for longer wavelength (~1300nm) QDs the thickness of the layers and surface patterns significantly increase allowing greatly reduced processing tolerances and applying further simplifications due to the possibility of using high contrast GaAs-AlOx DBRs.
An efficient numerical method for computing angle-resolved light scattering off periodic arrays is presented. The method combines finite-element discretization with a Schur complement solver. A significant speed-up of the computations in comparison to standard finite-element method computations is observed.
Rigorous optical simulations of 3-dimensional nano-photonic structures are an important tool in the analysis and optimization of scattering properties of nano-photonic devices or parameter reconstruction. To construct geometrically accurate models of complex structured nano-photonic devices the finite element method (FEM) is ideally suited due to its flexibility in the geometrical modeling and superior convergence properties. Reduced order models such as the reduced basis method (RBM) allow to construct self-adaptive, error-controlled, very low dimensional approximations for input-output relationships which can be evaluated orders of magnitude faster than the full model. This is advantageous in applications requiring the solution of Maxwell's equations for multiple parameters or a single parameter but in real time. We present a reduced basis method for 3D Maxwell's equations based on the finite element method which allows variations of geometric as well as material and frequency parameters. We demonstrate accuracy and efficiency of the method for a light scattering problem exhibiting a resonance in the electric field.
The finite-element method is a preferred numerical method when electromagnetic fields at high accuracy are to be computed in nano-optics design. Here, we demonstrate a finite-element method using hp-adaptivity on tetrahedral meshes for computation of electromagnetic fields in a device with rough textures. The method allows for efficient computations on meshes with strong variations in element sizes. This enables to use precise geometry resolution of the rough textures. Convergence to highly accurate results is observed.
Methods for solving Maxwell’s equations are integral part of optical metrology and computational lithography setups. Applications require accurate geometrical resolution, high numerical accuracy and/or low computation times. We present a finite-element based electromagnetic field solver relying on unstructured 3D meshes and adaptive hp-refinement. We apply the method for simulating light scattering off arrays of high aspect-ratio nano-posts and FinFETs.
We present novel numerical techniques for the simulation of the light outcoupling from state of the art organic light-emitting diodes (OLED). For the spatial discretization we use the finite element method which we apply in the frequency domain. To account for the large horizontal extension of the OLED we apply a recently proposed approach based on the Floquet transform which allows to restrict the calculations to the unit cell of a (quasi) periodic structure. Optically thick layers are efficiently treated by a plane wave expansion which we combine with the Finite Element Method by the domain decomposition method. We benchmark the new simulation tools for highly efficient state of the art OLED light extraction structures.
KEYWORDS: Glasses, Chemical elements, Finite element methods, Solar cells, Scattering, Electroluminescent displays, Light scattering, 3D modeling, Silicon, Multijunction solar cells
In many experimentally realized applications, e.g. photonic crystals, solar cells and light-emitting diodes, nanophotonic systems are coupled to a thick substrate layer, which in certain cases has to be included as a part of the optical system. The finite element method (FEM) yields rigorous, high accuracy solutions of full 3D vectorial Maxwell's equations1 and allows for great flexibility and accuracy in the geometrical modelling. Time-harmonic FEM solvers have been combined with Fourier methods in domain decomposition algorithms to compute coherent solutions of these coupled system.2, 3 The basic idea of a domain decomposition approach lies in a decomposition of the domain into smaller subdomains, separate calculations of the solutions and coupling of these solutions on adjacent subdomains. In experiments light sources are often not perfectly monochromatic and hence a comparision to simulation results might only be justified if the simulation results, which include interference patterns in the substrate, are spectrally averaged. In this contribution we present a scattering matrix domain decomposition algorithm for Maxwell's equations based on FEM. We study its convergence and advantages in the context of optical simulations of silicon thin film multi-junction solar cells. This allows for substrate lighttrapping to be included in optical simulations and leads to a more realistic estimation of light path enhancement factors in thin-film devices near the band edge.
KEYWORDS: Finite element methods, Electromagnetism, Computer simulations, Metrology, Stanford Linear Collider, Silver, Near field, Sensors, Numerical analysis, Chemical elements
Of keen interest to the IC industry are advanced computational lithography applications such as Optical Proximity Correction, OPC, Optical Proximity Effect matching, OPEM, and Source-Mask Optimization, SMO. Lithographic mask models used by these simulators and their interactions with scanner illuminator models are key drivers impacting the accuracy of the image predications of the computational lithography applications. To construct topographic mask model for hyper-NA scanner, the interactions of the fields with the mask topography have to be accounted for by numerically solving Maxwell’s equations. The simulators used to predict the image formation in the hyper-NA scanners have to rigorously treat the topographic masks and the interaction of the mask topography with the scanner illuminators. Such mask models come at a high computational cost and pose challenging accuracy vs. compute time tradeoffs. To address the high costs of the computational lithography for hyper-NA scanners, we have adopted Reduced Basis, RB, method to efficiently extract accurate, near field images from a modest sample of rigorous, Finite Element, FE, solutions of Maxwell’s equations for the topographic masks. The combination of RB and FE methods provides means to efficiently generate near filed images of the topographic masks illuminated at oblique angles representing complex illuminator designs. The RB method’s ability to provide reliable results from a small set of pre-computed, rigorous results provides potentially tremendous computational cost advantage. In this report we present RB/FE technique and discuss the accuracy vs. compute time tradeoffs of hyper-NA imaging models incorporating topographic mask images obtained with the RB/FE method. The examples we present are representative of the analysis of the optical proximity effects for the current generation of IC designs.
A method for automatic computation of parameter derivatives of numerically computed light scattering signals is demonstrated. The finite-element based method is validated in a numerical convergence study, and it is applied to investigate the sensitivity of a scatterometric setup with respect to geometrical parameters of the scattering target. The method can significantly improve numerical performance of design optimization, parameter reconstruction, sensitivity analysis, and other applications.
An overview on recent applications of the finite-element method Maxwell-solver JCMsuite to simulation tasks in nanooptics is given. Numerical achievements in the fields of optical metamaterials, plasmonics, photonic crystal fibers, light emitting devices, solar cells, optical lithography, optical metrology, integrated optics, and photonic crystals are summarized.
Nanostructures, like periodic arrays of scatters or low-index gratings, are used to improve the light outcoupling from organic light-emitting diodes (OLED). In order to optimize geometrical and material properties of such structures, simulations of the outcoupling process are very helpful. The finite element method is best suited for an accurate discretization of the geometry and the singular-like field profile within the structured layer and the emitting layer. However, a finite element simulation of the overall OLED stack is often beyond available computer resources. The main focus of this paper is the simulation of a single dipole source embedded into a twofold infinitely periodic OLED structure. To overcome the numerical burden we apply the Floquet transform, so that the computational domain reduces to the unit cell. The relevant outcoupling data are than gained by inverse Flouqet transforming. This step requires a careful numerical treatment as reported in this paper.
Photolithography simulations are widely used to predict, to analyze and to design imaging
processes in scanners used for IC manufacture. The success of these efforts is strongly dependent
on their ability to accurately capture the key drivers responsible for the image formation. Much
effort has been devoted to understanding the impacts of illuminator and projection lens models on
the accuracy of the lithography simulations [1-3]. However, of equal significance is the role of
the mask models and their interactions with the illuminator models.
We present a finite element method (FEM) solver for computation of optical resonance modes in VCSELs. We
perform a convergence study and demonstrate that high accuracies for 3D setups can be attained on standard
computers. We also demonstrate simulations of thero-optical effects in VCSELs.
Numerical simulations are an important tool for the design of opto-electronical components and devices. In
order to obtain realistic results, a multitude of physical effects and theories have to be included, e.g., Maxwell's
equations for lasing mode computations, heat transfer in active devices, and electronic transport. In our contribution
we perform coupled electro-thermal simulations of high power diode lasers. We analyze the temperature
dependence of the mode profile and far field characteristics. Our results will be compared to experimental
measurements of broad area lasers and will quantitatively describe the effect of thermal blooming.
EUV scatterometry is performed on 3D patterns on EUV lithography masks. Numerical simulations of the
experimental setup are performed using a rigorous Maxwell solver. Mask geometry is determined by minimizing
the difference between experimental results and numerical results for varied geometrical input parameters for
the simulations.
Optical metrology by scatterometry usually bases on the comparison of experimental and modeled light field
data. When solving inverse scatterometric problems, often not only a single simulation has to be carried out, but
multiple electromagnetic field solutions have to be computed for varying material and geometrical parameters
of the system under consideration. Then, high computational times for a single forward solution can make the
complete simulation task infeasible. Table based parameter reconstruction on the other hand has the disadvantage
of long offline computational times for creation of the library. Also an increasing number of variable parameters
can not be handled efficiently.
In this contribution we introduce the reduced basis method for creation of highly accurate reduced order
models of parametrized electromagnetic scattering problems. We apply our method to a real-world EUV metrology
application and show speed up factors of about 3000 in reconstruction time. Instead of several minutes or
hours EUV mask parameters can now be obtained in seconds, i.e., in real-time.
Comparison to direct microscopical measurements of the reconstructed geometry demonstrate the good performance
and maturity of our method.
Simulations of light scattering off an extreme ultraviolet lithography mask with a 2D-periodic absorber pattern
are presented. In a detailed convergence study it is shown that accurate results can be attained for relatively
large 3D computational domains and in the presence of sidewall-angles and corner-roundings.
Finite element methods (FEM) for the rigorous electromagnetic solution of Maxwell's equations are known to be
very accurate. They possess a high convergence rate for the determination of near field and far field quantities
of scattering and diffraction processes of light with structures having feature sizes in the range of the light
wavelength. We are using FEM software for 3D scatterometric diffraction calculations allowing the application
of a brilliant and extremely fast solution method: the reduced basis method (RBM). The RBM constructs a
reduced model of the scattering problem from precalculated snapshot solutions, guided self-adaptively by an error
estimator. Using RBM, we achieve an efficiency accuracy of about 10-4 compared to the direct problem with
only 35 precalculated snapshots being the reduced basis dimension. This speeds up the calculation of diffraction
amplitudes by a factor of about 1000 compared to the conventional solution of Maxwell's equations by FEM.
This allows us to reconstruct the three geometrical parameters of our phase grating from "measured" scattering
data in a 3D parameter manifold online in a minute having the full FEM accuracy available. Additionally, also
a sensitivity analysis or the choice of robust measuring strategies, for example, can be done online in a few
minutes.
KEYWORDS: Monte Carlo methods, Silicon, Absorption, Transparent conductors, Interfaces, Solar cells, Glasses, Chemical elements, Silicon films, Optical simulations
We apply a hybrid finite element / transfer matrix solver to calculate generation rate spectra of thin film
silicon solar cells with textured interfaces. Our focus lies on interfaces with statistical rough textures. Due to
limited computational domain size the treatment of such textures requires a Monte Carlo sampling of texture
representations to obtain a statistical average of integral target quantities. This contribution discusses our choice
of synthetic rough interface generation, the Monte Carlo sampling and the need for an incorporation of the cell's
substrate into optical simulation when illumination of the cell happens through the substrate. We present results
of the numerical characterization and generation rates for a single junction cell layout.
We present algorithmic details and applications of the reduced basis method as efficient Maxwell solver to
nanophotonic applications including examples from mask optimization in photolithography and parameter retrieval
in inverse problems, e.g., in optical metrology. The reduced basis method is a currently studied approach
to the multiple solution of problems depending on a number of geometrical, material and source parameters.
Such problems occur frequently in optimization tasks where parameters have to be adjusted in order to minimize
some error functionals or in production environments where deviations from ideal structures have to be
controlled.
Image modeling and simulation are critical to extending the limits of leading edge lithography technologies used
for IC making. Simultaneous source mask optimization (SMO) has become an important objective in the field of
computational lithography. SMO is considered essential to extending immersion lithography beyond the 45nm
node. However, SMO is computationally extremely challenging and time-consuming. The key challenges are due
to run time vs. accuracy tradeoffs of the imaging models used for the computational lithography.
We present a new technique to be incorporated in the SMO flow. This new approach is based on the reduced
basis method (RBM) applied to the simulation of light transmission through the lithography masks. It provides a
rigorous approximation to the exact lithographical problem, based on fully vectorial Maxwell's equations. Using
the reduced basis method, the optimization process is divided into an offline and an online steps. In the offline
step, a RBM model with variable geometrical parameters is built self-adaptively and using a Finite Element
(FEM) based solver. In the online step, the RBM model can be solved very fast for arbitrary illumination
and geometrical parameters, such as dimensions of OPC features, line widths, etc. This approach dramatically
reduces computational costs of the optimization procedure while providing accuracy superior to the approaches
involving simplified mask models. RBM furthermore provides rigorous error estimators, which assure the quality
and reliability of the reduced basis solutions.
We apply the reduced basis method to a 3D SMO example. We quantify performance, computational costs
and accuracy of our method.
Optical resonances in 1D photonic crystal microcavities are investigated numerically using finite-element light
scattering and eigenmode solvers. The results are validated by comparison to experimental and theoretical
findings from the literature. The influence of nanometer-scale geometry variations on the resonator performance
is studied. Limiting factors to ultra-high Q-factor performance are identified.
Optical properties of circular grating resonators in a silicon-on-insulator system are investigated numerically.
These structures are of interest as building-blocks of integrated photonic devices. We use a time-harmonic 3D
finite-element solver for studying transmission of waveguide modes through the system. We compare numerical
results to experimental findings from the literature.
Optical properties of hybrid plasmonic waveguides and of low-Q cavities, formed by waveguides of finite length
are investigated numerically. These structures are of interest as building-blocks of plasmon lasers. We use
a time-harmonic finite-element package including a propagation-mode solver, a resonance-mode solver and a
scattering solver for studying various properties of the system. Numerical convergence of all used methods is
demonstrated.
A bottleneck for computational lithography and optical metrology are long computational times for near field
simulations. For design, optimization, and inverse scatterometry usually the same basic layout has to be simulated
multiple times for different values of geometrical parameters.
The reduced basis method allows to split up the solution process of a parameterized model into an expensive
offline and a cheap online part. After constructing the reduced basis offline, the reduced model can be solved
online very fast in the order of seconds or below. Error estimators assure the reliability of the reduced basis
solution and are used for self adaptive construction of the reduced system.
We explain the idea of reduced basis and use the finite element solver JCMsuite constructing the reduced
basis system. We present a 3D optimization application from optical proximity correction (OPC).
Light transmission through a 2D-periodic array of small rectangular apertures in a film of highly conductive
material is simulated using a finite-element method. It is demonstrated that well converged results are obtained
using higher-order finite-elements. The influence of the array periodicity and of corner roundings on transmission
properties is investigated.
Light transmission through circular subwavelength apertures in metallic films with surrounding nanostructures
is investigated numerically. Numerical results are obtained with a frequency-domain finite-element method.
Convergence of the obtained observables to very low levels of numerical error is demonstrated. Very good
agreement to experimental results from the literature is reached, and the utility of the method is demonstrated
in the investigation of the influence of geometrical parameters on enhanced transmission through the apertures.
We have developed an interface which allows to perform rigorous electromagnetic field (EMF) simulations with
the simulator JCMsuite and subsequent aerial imaging and resist simulations with the simulator Dr.LiTHO.With
the combined tools we investigate the convergence of near-field and far-field results for different DUV masks.
We also benchmark results obtained with the waveguide-method EMF solver included in Dr.LiTHO and with
the finite-element-method EMF solver JCMsuite. We demonstrate results on convergence for dense and isolated
hole arrays, for masks including diagonal structures, and for a large 3D mask pattern of lateral size 10 microns
by 10 microns.
KEYWORDS: Scattering, Maxwell's equations, Photomasks, Radio propagation, Waveguides, Chemical elements, Scatterometry, Magnetism, Electromagnetism, Systems modeling
Rigorous electromagnetic field simulations are an essential part for scatterometry and mask pattern design.
Today mainly periodic structures are considered in simulations. Non-periodic structures are typically modeled
by large, artificially periodified computational domains. For systems with a large radius of influence this leads to
very large computational domains to keep the error sufficiently small. In this paper we review recent advances
in the rigorous simulation of isolated structures embedded into a surrounding media. We especially address the
situation of a layered surrounding media (mask or wafer) with additional infinite inhomogeneities such as resist
lines. Further we detail how to extract the far field information needed for the aerial image computation in the
non-periodic setting.
Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical
metrology and design of nanostructured optical components. A vectorial finite element method (FEM) is a good
choice for an accurate modeling of complicated geometrical features. However, from a numerical point of view
solving the arising system of linear equations is very demanding even for medium sized 3D domains. In numerics,
a domain decomposition method is a commonly used strategy to overcome this problem. Within this approach
the overall computational domain is split up into smaller domains and interface conditions are used to assure
continuity of the electromagnetic field. Unfortunately, standard implementations of the domain decomposition
method as developed for electrostatic problems are not appropriate for wave propagation problems. In an earlier
paper we therefore proposed a domain decomposition method adapted to electromagnetic field wave propagation
problems. In this paper we apply this method to 3D mask simulation.
We discuss realization, properties and performance of the adaptive finite element approach to the design of optical
waveguides. Central issues like the construction of higher-order vectorial finite elements, local error estimation,
automatic and adaptive grid refinement, transparent boundary conditions and fast linear system solution by
domain decomposition techniques will be discussed.
We present rigorous 3D EMF simulations of isolated features on photomasks using a newly developed finite-element
method. We report on the current status of the finite-element solver JCMsuite, incorporating higher-order
edge elements, adaptive refinement methods, and fast solution algorithms. We demonstrate that rigorous
and accurate results on light scattering off isolated features can be achived at relatively low computational cost,
compared to the standard approach of simulations on large-pitch, periodic computational domains.
Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical
metrology and optics design of nanostructured components. As has been shown in previous benchmarks some of
the presently used methods suffer from low convergence rates and/or low accuracy of the results and exhibit very
long computation times1, 2 which makes application to extended 2D layout patterns impractical. We address 3D
simulation tasks by using a finite-element solver which has been shown to be superior to competing methods by
several orders of magnitude in accuracy and computational time for typical microlithography simulations.2 We
report on the current status of the solver, incorporating higher order edge elements, adaptive refinement methods,
and fast solution algorithms. Further, we investigate the performance of the solver in the 3D simulation project
of light diffraction off an alternating phase-shift contact-hole mask.
Adaptive finite elements are the method of choice for accurate simulations of optical components. However as
shown recently by Bienstman et al. many finite element mode solvers fail to compute the propagation constant's
imaginary part of a leaky waveguide with sufficient accuracy. In this paper we show that with a special goal
oriented error estimator for capturing radiation losses this problem is overcome.
Hollow-core holey fibers are promising candidates for low-loss guidance of light in various applications, e.g., for the use in laser guide star adaptive optics systems in optical astronomy. We present an accurate and fast method for the computation of light modes in arbitrarily shaped waveguides. Maxwell's equations are discretized using vectorial finite elements (FEM). We discuss how we utilize concepts like adaptive grid refinement, higher-order finite elements, and transparent boundary conditions for the computation of leaky modes in photonic crystal fibers. Further, we investigate the convergence behavior of our methods. We employ our FEM solver to design hollow-core photonic crystal fibers (HCPCF) whose cores are formed from 19 omitted cladding unit cells. We optimize the fiber geometry for minimal attenuation using multidimensional optimization taking into account radiation loss (leaky modes).
We present rigorous simulations of EUV masks with technological imperfections like side-wall angles and corner roundings. We perform an optimization of two different geometrical parameters in order to fit the numerical results to results obtained from experimental scatterometry measurements. For the numerical simulations we use an adaptive finite element approach on irregular meshes. This gives us the opportunity to model geometrical structures accurately. Moreover we comment on the use of domain decomposition techniques for EUV mask simulations. Geometric mask parameters have a great influence on the diffraction pattern. We show that using accurate simulation tools it is possible to deduce the relevant geometrical parameters of EUV masks from scatterometry measurements. This work results from a collaboration between AMTC (mask fabrication), Physikalisch-Technische Bundesanstalt (scatterometry) and ZIB/JCMwave (numerical simulation).
We perform 3D lithography simulations by using a finite-element
solver.
To proof applicability to real 3D problems we investigate
DUV light propagation through a structure of size 9μm x 4μm x 65nm.
On this relatively large computational domain we
perform rigorous computations (No Hopkins) taking into account
a grid of 11 x 21 source points with two polarization directions
each.
We obtain well converged results with an accuracy of the
diffraction orders of about 1%.
The results compare well to experimental aerial imaging results.
We further investigate the convergence of 3D solutions towards
quasi-exact results obtained with different methods.
Rigorous numerical simulations of Maxwell's equations are extremely helpful in the understanding of physical effects in nano-optics and are essential for the design of nano-optical devices. We have developed a finite-element (FEM) package for the solution of eigenvalue and scattering problems resulting from Maxwell's equations. We have implemented higher order vectorial elements, adaptive mesh refinement, transparent boundary conditions based on the Pole condition, and fast algorithms. In this contribution we discuss the application of the FEM solvers to recent experiments in nano-photonics.
Miniaturized optical resonators with spatial dimensions of the order of the wavelength of the trapped light offer prospects for a variety of new applications like quantum processing or construction of meta-materials. Light propagation in these structures is modelled by Maxwell's equations. For a deeper numerical analysis one may compute the scattered field when the structure is illuminated or one may compute the resonances of the structure. We therefore address in this paper the electromagnetic scattering problem as well as the computation of resonances in an open system. For the simulation effcient and reliable numerical methods are required which cope with the infinite domain. We use transparent boundary conditions based on the Perfectly Matched Layer Method (PML) combined with a novel adaptive strategy to determine optimal discretization parameters like the thickness of the sponge layer or the mesh width. Further a novel iterative solver for time-harmonic Maxwell's equations is presented.
An extremely fast time-harmonic finite element solver developed for the transmission analysis of photonic crystals was applied to mask simulation problems. The applicability was proven by examining a set of typical problems and by a benchmarking against two established methods (FDTD and a differential method) and an analytical example. The new finite element approach was up to 100 times faster than the competing approaches for moderate target accuracies, and it was the only method which allowed to reach high target accuracies.
KEYWORDS: Magnetism, Split ring resonators, Geometrical optics, Near infrared, Light scattering, Chemical elements, Metamaterials, Finite element methods, Polarization, Gold
It seems to be feasible in the near future to exploit the properties of left-handed metamaterials in the telecom or even in the optical regime. Recently, split ring-resonators (SRR's) have been realized experimentally in the near infrared (NIR) and optical regime.1, 2 In this contribution we numerically investigate light propagation through an array of metallic SRR's in the NIR and optical regime and compare our results to experimental results. We find numerical solutions to the time-harmonic Maxwell's equations by using advanced finite-element-methods (FEM). The geometry of the problem is discretized with unstructured tetrahedral meshes. Higher order, vectorial elements (edge elements) are used as ansatz functions. Transparent boundary conditions (a modified PML method3) and periodic boundary conditions4 are implemented, which allow to treat light scattering problems off periodic structures.
This simulation tool enables us to obtain transmission and reflection spectra of plane waves which are incident onto the SRR array under arbitrary angles of incidence, with arbitrary polarization, and with arbitrary wavelength-dependencies of the permittivity tensor. We compare the computed spectra to experimental results and investigate resonances of the system.
We present our simulation tool JCMmode for calculating propagating modes of an optical waveguide. As ansatz functions we use higher order, vectorial elements (Nedelec elements, edge elements). Further we construct transparent boundary conditions to deal with leaky modes even for problems with inhomogeneous exterior domains as for integrated hollow core Arrow waveguides. We have implemented an error estimator which steers the adaptive mesh refinement. This allows the precise computation of singularities near the metal's corner of a Plasmon-Polariton waveguide even for irregular shaped metal films on a standard personal computer.
We present a finite-element simulation tool for calculating
light fields in 3D nano-optical devices. This allows to solve challenging problems on a standard personal computer. We present solutions to eigenvalue problems, like Bloch-type eigenvalues in photonic crystals and photonic crystal waveguides, and to scattering problems, like the transmission through finite photonic crystals.
The discretization is based on unstructured tetrahedral grids with
an adaptive grid refinement controlled and steered by an error-estimator. As ansatz functions we use higher order, vectorial elements (Nedelec, edge elements). For a fast convergence of the solution we make use of advanced multi-grid algorithms adapted for the vectorial Maxwell's equations.
We review key algorithms for the numerical solution of waveguide eigenvalue problems and discuss their application to a typical simulation problem in integrated optics - the computation of eigenmodes of a MQW laser structure. Here we focus mainly on a self adaptive realization of the codes supplying solutions with prescribed
accuracy in a CPU-time as short as possible. In a brief outlook we show how to extend these principles to solve general time-harmonic and time-dependent scattering problems in an adaptive finite element context.
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